Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence
Propagating Ranking Functions on a Graph: Algorithms and Applications
Buyue Qian
Xiang Wang
Ian Davidson
IBM T. J. Watson Research
Yorktown Heights, NY 10598
bqian@us.ibm.com
IBM T. J. Watson Research
Yorktown Heights, NY 10598
wangxi@us.ibm.com
University of California
Davis, Davis, CA 95616
davidson@cs.ucdavis.edu
Abstract
methods to estimate the unknown or improve the weak ranking functions by those strong ones. The purpose of this work
is to propagate linear ranking functions over the nodes on a
graph, where each node corresponds to a retrieval task/user.
Problem Setting. There are three key concepts in this
work, (i) linear ranking function, (ii) ordinal distribution features, and (iii) ranking function propagation. We shall now
explain each briefly. A linear ranking function is a realvalued vector, whose inner product with a data point is usually called a ranking score. In a retrieval task using a linear
ranking function, we first calculate ranking scores, the inner product between the ranking function vector and all data
points, and then sort the data points based on the ranking
scores. Data points with higher ranking scores are more relevant to the retrieval topic. Ordinal distribution features are a
powerful way of representing complex objects by the counts
of features that are ordinal, that is the features represent an
ordered set of values such as small, medium, large.
Consider our experiments on creating a ranking function for
each type of bird based on their song. The songs are represented using spectrograms which contains the counts of each
audio frequency and analogous to histograms. The spectrograms capture not only the counts of each frequency but also
that adjacent (in the spectrogram) frequencies are similar.
Propagation of ranking functions implies that in our study a
graph can be comprised of strong, weak, and unknown ranking functions, and we allow the strong ranking functions being propagated to the weak or unknown functions. In particular we can calculate the ranking functions after an infinite
number of propagation steps on the network.
Proposal. To explore the graph propagation of linear
ranking functions on distribution data, there are three main
challenges that we address. (i) Since our ranking function
is of ordinal (order sensitive features) regular label propagation methods will fare poorly with these ordinal distribution features. We adopt the Hellinger distance metric to
calculate the similarity between two linear ranking functions (distributions). (ii) Propagating distributed based ranking functions requires a specialized objective function. We
generalize the objective functions of several classic diffusion methods whose objectives have shown to be useful in
solving real problems. (iii) The construction of a similarity
graph to propagate the ranking functions will vary between
applications. For example, in a personalized movie ranking
Learning to rank is an emerging learning task that opens
up a diverse set of applications. However, most existing work focuses on learning a single ranking function whilst in many real world applications, there can
be many ranking functions to fulfill various retrieval
tasks on the same data set. How to train many ranking functions is challenging due to the limited availability of training data which is further compounded
when plentiful training data is available for a small subset of the ranking functions. This is particularly true in
settings, such as personalized ranking/retrieval, where
each person requires a unique ranking function according to their preference, but only the functions of the persons who provide sufficient ratings (of objects, such as
movies and music) can be well trained. To address this,
we propose to construct a graph where each node corresponds to a retrieval task, and then propagate ranking functions on the graph. We illustrate the usefulness
of the idea of propagating ranking functions and our
method by exploring two real world applications.
Introduction
Any system that presents ordered results to users is performing ranking (Burges et al. 2005). This has led to extensive interest in applying machine learning techniques to learn ranking functions. Such techniques are called learning to rank,
whose goal is to automatically build a retrieval function from
training data, and has been successfully applied to various
information retrieval problems. In many real world applications, we often require many ranking functions to fulfill
various retrieval tasks on a single dataset. It is often the case
that the training data is limited and imbalanced hence, only
a few ranking functions can be trained adequately (deemed
as strong), while the majority ranking functions are completely unknown or learnt with little training data (deemed
as weak). Consider a personalized movie ranking problem,
each person has different taste of movies and thus requires
a unique ranking function. However, for a particular user, a
strong ranking function can be learned only if that user provides sufficient movie ratings, which is often not the case
in practice. In this study we consider using graph diffusion
c 2015, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
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of features matters to learning algorithms. The latter type
of ordinal distribution features is commonly used in learning
to rank problems and also covers a large portion of data representations, such as the color histogram used in computer
vision, the spectrogram used in natural language processing, and various transformations used in signal processing.
While on the data of independent features, ranking functions
can be readily propagated using typical label propagation
methods, there is no available method, to our knowledge, to
perform ranking function propagation on distribution data.
Existing graph diffusion methods perform propagation in a
bin-by-bin fashion and ignore the interactions between adjacent histogram bins. We shall in the following example show
that sometimes shifting and preserving shape are preferred.
problem, the graph can be the user co-rating matrix or constructed using side information such as user profiles.
Though the goals look similar, our work differs from
multi-task learning (MTL) in two aspects. (i) MTL usually
assumes just a few tasks whilst our method performs propagation on a large set of ranking functions (2000+ in one of
our experiments). (ii) While MTL tries to learn a set of cooperative tasks together, our method views the learning of the
initial set of ranking functions as a preprocessing step, and
only focuses on the inference of unknown ranking functions.
Contribution and Claims. Our work makes the following three technical contributions. (i) We explore a new
method to efficiently learn thousands of ranking functions
from insufficient and imbalanced training data. (ii) We propose a generic diffusion method to propagate linear ranking functions modeled as distributions on a graph. Our approach allows for interactions between adjacent histogram
bins making propagation more useful. (iii) The proposed
method can be naturally deployed to network settings, such
as personalized ranking and social network advertising.
(a)
Related Work
Our work relates to two topics: label propagation and learning to rank. We briefly review related work in both areas.
Graph-based label propagation mainly aims to address
the issue of transduction. Blum and Chawla have proposed
label propagation methods on graphs as a mincut problem.
The Gaussian fields and harmonic function (GFHF) method
in (2003) is a continuous relaxation to the difficult discrete Markov random fields. The local and global consistency (LGC) method (Zhou et al. 2003) extends GFHF with
the normalized Laplacian. Another manifold regularization
framework (2006) employs two regularization terms, i.e., a
base kernel and a L2 regularizer. Lawrence and Jordan propose to learn with unlabeled data in the context of Gaussian
process. Comprehensive surveys on semi-supervised learning can be found in (Zhu 2005) and (Chapelle et al. 2006).
Learning to rank algorithms fall into three categories
which differ in the form of training data. (i) Pointwise methods approximate a ranking problem as ordinal regression
with large margin formulations (Shashua and Levin 2002)
(Crammer and Singer 2001), and subset ranking formulations(Cossock and Zhang 2006). (ii) Pairwise methods learn
from pairs of instances ordered by their relevance to the
ranking problem (iii) Listwise use a fully ordered rank list
as an instance, e.g., ListNet (Cao et al. 2007), AdaRank (Xu
and Li 2007) and SVM Map (Yue et al. 2007). Learning to
rank approaches have shown to be useful in many information retrieval applications. RankSVM was applied Joachims
to document retrieval, RankNet (Burges et al. 2005) has
shown to be useful on large scale web search. A brief survey of learning to rank approaches exists (Hang 2011).
A graph with each node correspondes to a distribution/ranking
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minus a constant
Figure 1: Distribution propagation using different methods.
Note that Figure 1(h) is equivalent to Figure 1(g), since each
distribution here is viewed as a linear ranking function
To illustrate the importance of distribution propagation,
let us consider a toy graph with only three nodes. Figure 1(a)
presents a simple graph with three nodes, each of which corresponds to a distribution (linear ranking function). We see
that in the given Distribution 1 (Figure 1(b)) there are three
bars on the left side of the 5-bin histogram with the middle
bar higher than the other two, and the Distribution 2 (Figure 1(d)) follows the same shape as Distribution 1 but locates on the right side of the 5-bin histogram. It is clear that
shifting Distribution 1 to the right by two bins is equivalent
to Distribution 2, and vice versa. Therefore, we can infer the
ideal propagation on distribution data as in Figure 1(c).
We next infer the missing distribution (in the middle) using different graph propagation methods. Figure 1(e) and
1(f) show the inferred distributions using two classic graph
The Importance of Propagating Distributions
Feature representations of data objects fall into two categories. (i) Independent features, where the ordering of features is permutable without affecting the learning result. (ii)
Ordinal distribution/histogram features, where the ordering
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Sij in the matrix S indicates the similarity between the retrieval tasks ti and tj , Sij = 1 if ti and tj are exactly the
same and Sij = 0 if ti and tj are completely irrelevant, and
we have Sij = Sji . Let D denote the degree
Pm matrix of S,
which is a diagonal matrix and Dii =
j=1 Sij . Before
propagating ranking functions on graph, we are given a initial set (possibly incomplete) of ranking functions obtained
from some learning to rank algorithm, which is denoted using Y , Y ∈ Rm×d , and Yi· (i-th row of Y ) is the given
ranking function of retrieval task ti . We set Yi· to a vector
comprised of zeros if the ranking function for ti is missing.
The target value in our formulation is the propagation result
of ranking functions, which is denoted by W , W ∈ Rm×d ,
and Wi· (i-th row of W ) is the resulting (propagated) ranking function for retrieval task ti .
We assume the initial set of ranking functions Y consists of a small portion of strong (sufficiently trained) ranking functions and a large portion of weak/unknown (insufficiently/not trained) ranking functions. Our goal is to fill
in the unknown or improve the weak ranking functions using the strong functions based on the similarity amongst retrieval tasks, i.e., to propagate the ranking functions in Y
via the similarity matrix S of the graph. We follow the classic consistency assumption of graph-based semi-supervised
learning, nearby ranking functions are likely to be similar,
and generalize the objective functions from previous studies.
Formally, the generalized objective is written as follows.
diffusion methods, GFHF and LGC, respectively. We observe that traditional methods do not work well on distribution data, since their result is essentially a weighted average of the given distributions. Figure 1(g) shows the result
of our method, which captured the key structure of the two
given distributions (higher in the middle and lower on both
sides). Since our study concerns ranking function propagation, each distribution here can be viewed as a linear ranking
function. As adding a constant to a linear ranking function
on distribution data (nonnegative and sum-to-one) does not
change the ranking order, we can remove the redundant constant and have an equivalent ranking function as shown in
Figure 1(h). The result is very close to the ideal propagation.
The Method
Preliminaries
Formally, the problem to address is described as follows.
In a retrieval problem, we are given a dataset containing
n instances X = {x1 , x2 , · · · , xn }, which are represented
as distributions in Rd space. Since ranking functions of
non-distribution data can be readily handled by traditional
graph propagation methods, this paper only concerns ordinal distribution based feature representations, i.e., we assume the features of data are (i) nonnegative, (ii) sumto-one, and (iii) the order of features is not permutable.
Such representations have been shown to be useful for ranking problems of images (Datta et al. 2008). On the set of
instances X , there are m ranking tasks/categories T =
{t1 , t2 , · · · , tm }, which correspond to a set of ranking functions W = {w1 , w2 , · · · , wm } respectively. Each ranking
function is defined with respect to an external ordering (retrieval task) that we are trying to learn from the data. Since
in our approach we adopt a linear ranking function, i.e. wi is
a real-valued vector with the same dimension as xi , the ranking score of an instance xi w.r.t. a retrieval tasks tj is simply
calculated from the inner product between xi and wj .
γij = wjT xi
arg min
W
m
X
i,j=1
Sij d2 (Wi· , Wj· ) +
m
X
µi d2 (Wi· , Yi· )
(2)
i=1
where d() denotes some distance metric, and d2 () simply
denotes the square of a distance. The first term measures
the smoothness of graph, i.e., two similar ranking functions
should have small distance. The second term penalizes any
changes to the initial ranking functions, i.e., the given strong
ranking functions should not be overwritten significantly
since these functions are deemed as correct. µ is a tuning
parameter that balances the smoothness and penalty. In practice, µ can be a m-length vector, such that we can set a large
overwriting penalty to the locations of strong ranking functions, and a zero/small penalty for the overwriting of unknown/weak ranking functions respectively. We shall later in
this paper experimentally demonstrate that the performance
of our proposed approach is not sensitive to the scale of µ.
The generalized objective is closely related to several
classic graph-based semi-supervised learning methods. If
d(Wi· , Wj· ) = |Wi· − Wj· | the objective reduces to GFHF
W
(graph Laplacian). If d(Wi· , Wj· ) = | √WDi· − √ j· | the ob-
(1)
We construct a graph over ranking functions using a measure of similarity that is application dependent and allow
the interactions amongst themselves. In particular, we assume that among the m ranking functions, there are only a
small portion of ranking functions that are trained with sufficient data (deemed as strong functions), and the majority
remaining ones are unknown or learned with little training
data (deemed as unknown or weak functions). In this paper
we present a general approach to propagate linear ranking
functions, and the input of our method is a set of ranking
functions. Any learning to rank methods that produce linear
ranking functions can be served as a preprocessing component for our approach.
ii
Djj
jective reduces to LGC (normalized graph Laplacian).
With the objective function, the problem turns to selecting
an appropriate distance metric. Though propagation on nondistribution data can be naturally handled by typical graph
diffusion approaches, the propagation of distributions is difficult to solve. Similar problems involving probability interpolation have been encountered in mathematical literatures
(McCann 1997; Agueh and Carlier 2011), topic modeling
(Cai, Wang, and He 2009), clustering problem (Applegate
Objective Function
To perform ranking function propagation, we first construct
a graph of ranking functions, which is defined by a symmetric similarity/affinity matrix S, S ∈ Rm×m . Note that
the graph contains both strong and unknown/weak ranking
functions to enable the propagation of functions. An entry
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ranking functions to be inferred. To recover W in closed
form, we write the objective in Eq.(4) into matrix format.
et al. 2011), and also graphics applications (Bonneel et al.
2011). To address this issue, our algorithm was designed
with a few criteria in mind. (1) The distance metric is preferred to be within the range from 0 to 1. (2) It allows interactions between adjacent histogram bins. (3) From the distance metric, we should be able to derive a simple or closedform solution for the objective. A comparison of statistical
distances found the Hellinger distance metric satisfies all the
aforementioned criteria. The Hellinger distance, a type of f divergence, is used to quantify the similarity between two
probability distributions. Let a and b denote two distributions, the Hellinger metric in discrete format is defined as
s
X √
p 2
1
H(a, b) = √
ai − bi
(3)
2
i
Q(W )
W
1
∂Q
1
∂W ◦ 2
1
2
i=1
µi
d
X
√
wij −
√ 2
yij
1
1
(6)
where I denotes the identity matrix, and ()◦2 denotes the
element-wise square of a matrix. The propagation of ranking
functions is closely related to random walk on a graph, but
here the strong ranking functions are viewed as “absorbing
boundary” for the random walk. Our method differs from
typical random walk in two main aspects, (i) we penalize the
changes on the strong ranking functions, and (ii) our solution
is an equilibrium state in terms of hitting time.
k=1
+
1
= (D − S)W ◦ 2 + µ(W ◦ 2 − Y ◦ 2 ) = 0
With simple linear algebra we have the global optimal W .
!◦2
−1
1
1
(7)
(D − S) + I
Y ◦2
W =
µ
m
d
X
2
√
1 X
√
Sij
wik − wjk
2 i,j=1
m
X
1T
1
1
tr{ W ◦ 2 (D − S)W ◦ 2
2
1
1
1
1
1
+ µ(W ◦ 2 − Y ◦ 2 )T (W ◦ 2 − Y ◦ 2 )} (5)
2
where tr{} denotes the trace operation, and ()◦ 2 the denotes
element-wise square root of a matrix.
1
Since W ◦ 2 and W have one-to-one correspondence, the
optimization of Q with respect to W is equivalent to min1
1
imizing Q with respect to W ◦ 2 . Since W ◦ 2 is continuous
and our objective function is convex, we recover the mini1
mum by simply setting the derivative w.r.t. W ◦ 2 to zero.
In addition there are other benefits brought by the
Hellinger metric: (4) Unlike other statistical distances, the
Hellinger metric is applicable to any distributions without
pre-processing corrections, e.g., KL divergence prohibits zeros on the same location of both distribution which is corrected using the Laplace correction, (5) The Hellinger distance has shown to be robust to noise and skew-insensitive to
imbalanced data (Cieslak et al. 2012; Goldberg et al. 2009).
Substituting the Hellinger metric into the Eq.(2), our objective function is written as
arg min
=
(4)
j=1
Empirical Evaluation
From the definition of the Hellinger distance metric, we
see that it is required that all entries in a linear ranking function are (i) non-negative (wij ≥ 0, ∀i, j) in order to keep the
metric valid, and the entries in a function are (ii) sum-to-one
Pd
( j=1 wij = 1, ∀i) so as to preserve the nice properties of
the Hellinger metric. However, in practice these two requirements rarely hold in a linear ranking function learned by
some learn to rank algorithm. To address this issue, we adopt
a simple scheme to preprocess linear ranking functions: (1)
adding/subtracting a constant to make the entries range from
zero to a positive number, and then (2) multiplying/dividing
a constants so that the entries sum to one. Since our work
only deals with distribution data, the above two manipulations do not affect the ranking orders. Recall that a ranking
score is calculated using rij = wjT xi . For adding/subtracting
a constant we have rij = (wj + c)T xi = wjT xi + c, since
Pd
j=1 wij = 1. As for multiplying/dividing a constant, we
have rij = cwjT xi . We see that both the shifting and normalization make an uniform change to all ranking scores,
therefore, the ranking orders remain exactly the same.
We in this section attempt to understand the strengths and
relative performance of our distribution based ranking function propagation method, which we refer to in this section
as RFP (Ranking Function Propagation). It is important to
note that, to the best of our knowledge, our work is the
first on ranking function propagation on distribution data.
Therefore, we in our experiment can only compare with
non-distribution propagation methods. In particular we wish
to answer how well our method compares to the following
state-of-the-art baseline methods:
1. GFHF (2003): A state-of-the-art label propagation method
based on the harmonic function and the graph Laplacian.
2. LGC (2003): A label propagation method based on the
graph smoothness and the normalized graph Laplacian.
The result shows that our propagation method significantly outperforms the two baseline methods. Given this, the
next question naturally raised would be “How good are the
inferred ranking functions comparing to the ranking functions that are trained with sufficient data?”. To investigate
this we explore the following two extreme scenarios:
3. Lower Bound: performance of the initial ranking functions without propagation. For unknown ranking functions, we simply use a random vector. An inferred ranking
function should be at least better than a random guess.
Closed-form Solution
In the proposed learning objective, the cost function involves
only one variable W to be optimized, which contains the
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4. Upper Bound: performance of all the ranking functions
fully trained, which can be deemed as 1−training errors,
and is the upper bound for any propagation methods.
recordings, such as Brown Creeper, Pacific Wren. Each audio recording is paired with a set of species that are present.
These species labels were obtained by listening to the audio
and looking at spectrograms. Our goal is to learn 19 ranking
functions to retrieve the 19 species of bird. We first construct
a graph of the 19 nodes (species). The affinity matrix S is
constructed by calculating the Hellinger similarity (i.e., one
minus the Hellinger distance) amongst the prototypes (averaged audio spectrogram of a species) of the 19 bird species.
We then propagate ranking functions on the graph.
Learning to rank model. We use RankSVM (Chapelle
2007) to provide the initial ranking functions for the propagation methods. The pairwise constraints used for training
in RankSVM are generated using labels (in the bird species
dataset) or the ratings (in the movie poster dataset). The parameters in RankSVM are set as follows: the penalty constant C is set to 1, and the two options for Newton’s method,
i.e., the maximum number of linear conjugate gradients is
20, the stopping criterion for conjugate gradients is 10−3 .
Parameters. GFHF is nonparametric. For LGC and RFP,
there is a parameter µ that balances the graph smoothness
and overwriting penalty. In LGC, µ is set using cross validation. In RFP, we set the vector µ with the following values, 10 at the locations corresponding to strong (trained with
abundant data) ranking functions, 0 at the locations corresponding to unknown (no training data) ranking functions,
and 1 at the locations corresponding to weak (trained with
little data) functions. We will also experimentally show that
our propagation method is not sensitive to the scale of µ.
Evaluation measure. In our evaluation, ranking accuracy
was assessed using a standard measure - normalized discounted cumulative gain (NDCG). We use binary ratings to
calculate NDCG, i.e., rating is 1 if relevant, and 0 otherwise.
To further investigate the properties of inferred ranking functions, since a linear ranking function can be viewed as a vector in the space, we calculate the cosine similarity between
an inferred ranking function and the corresponding ranking
functions that is fully trained using all data (upper bound).
(a)
Average NDCG at top 20
(c)
Experiment 1. Acoustic Retrieval of Bird Species
(b)
Cosine similarity between inferred
ranking functions and the fully trained
(deemed as the ground truth) functions
Average NDCG w.r.t. different µ
Figure 3: Evaluation on birds dataset (standard deviation is denoted by shade).
Result and discussion. In each trial we randomly select 5
species of birds, and train a strong ranking function for each
of them using adequate training data. The 5 strong ranking
functions are used as the initial set of ranking functions,
and then additional 150 (randomly selected) labeled audio
recordings are gradually added to the training set, which are
used to build new ranking functions, or improve the existing weak ones. At each step, we infer the unknown or weak
ranking functions using the five graph propagation methods.
The experiment is repeated for 100 times.
The mean (averaged over both the 19 retrieval tasks and
100 random trials) NDCG scores (at rank 20) are reported in
Figure 3(a). We see that both the GFHF and LGC achieve
significantly higher accuracy than Lower bound. This
confirms the motivation and usefulness of ranking function
propagation since it improves the retrieval performance with
no additional training data. It can be observed that our distribution propagation method significantly outperforms the
competing techniques, and approaches the Upper bound
as the training set increases. Figure 3(b) shows the cosine
similarity between the inferred ranking functions and the
ranking functions that are fully trained with all data (deemed
as the ground truth). The result also verifies the superior performance of our method. This demonstrates the effective-
Figure 2: Song meter data collection locations in the H. J. Andrews Forest.
Datasets and Experimental Settings. The birds dataset
(Briggs et al. 2013) consists of 645 ten-second audio recordings collected from the Cascade mountain range of Oregon (as shown in Figure 2). The raw audio signal is converted into a spectrogram (an image representation of the
sound), by dividing it into frames, and applying the FFT to
each frame. To extract features, the spectrogram is divided
into 24 frequency bands, and we then summarize each band
with statistics, including the mean, variance, skewness, kurtosis, min, max, and median. Finally, each audio recoding
is represented using a 168-dimensional statistical spectrogram descriptor (SSD), which can be viewed as distribution
data. There are 19 species of bird presented in the audio
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ness and necessity of the proposed graph diffusion method
since the propagation of distributions differs from traditional
propagation methods that were derived to propagate vectors
of independent labels. By observing the performance comparison of our method w.r.t different values of µ shown in
Figure 3(c), we can conclude that our method is not sensitive to the scale of parameter µ, since the ranking accuracy is
about the same even when µ is significantly different (setting
Our-mu=1 denotes µ = 1 for strong ranking functions and
µ = 0.1 for weak ones, Our-mu=10 is ten times bigger).
each step, we perform ranking functions propagation on the
user graph, and the experiment is repeated for 50 times.
Experiment 2. Personalized Movie Ranking
(a)
Average NDCG at top 200
(b)
Cosine similarity between inferred
ranking functions and the fully trained
(deemed as the ground truth) functions
Figure 4: Movie posters crawled from the web
Datasets and Experimental Settings. The dataset used
in this experiment is a collection of movie posters,
which are crawled from the web using the links provided in HetRec2011-MovieLens-2K dataset (Cantador,
Brusilovsky, and Kuflik 2011). Note that the goal of this experiment is not to evaluate our method as a recommender
system, rather just to show the capability of our method in
content-based personalized ranking. The HetRec-2K dataset
contains 855,598 personal ratings on 10,197 movies from
2,113 users. On average, there are 404.921 ratings per user,
and 84.637 ratings per movie. We explore an interesting
scenario of personalized movie ranking – rank movies for
each user based on the movie posters. This is inspired by
a recent online article1 , which suggests that movie posters,
as people’s first impression of a movie, are closely correlated to people’s taste of movies. Using the picture URLs
provided in the dataset, we collected 9,893 movie posters
from IMDb website (we will make the movie poster dataset
publicly available soon). We calculate the color histogram
(4 × 4 × 4 bins), such that each movie poster is represented
by a 64-dimensional vector. According to the definition of
color histogram, a representation of the distribution of colors
in an image, the data can be viewed as distribution data. The
9,893 movie posters we crawled are associated with 835,189
ratings from 2,112 users. Our goal is to build a distinct ranking function for each user. The user affinity matrix S is constructed by simply normalizing the co-rating matrix of users,
which consists of the counts that both users rated the same
movies. In our evaluation, the user ratings (ranged from 0.5
to 5) are used to calculate the NDCG scores.
(c)
The mean NDCG scores at rank 200, averaged over the
2,112 users and 50 random trials, are reported in Figure 5(a).
It can be observed that the three propagation methods (i.e.,
RFP, GFHF, and LGC) significantly outperform the Lower
bound baseline. This confirms our motivation of ranking
function propagation, as it does increase the ranking accuracy without extra training data. Among the three propagation methods, we see that our distribution propagation
achieves the highest ranking accuracy. The cosine similarity
between the inferred ranking functions and the ground truth
is presented in Figure 5(b), which shows that the ranking
functions learned from our method are closer to the ground
truth than the two competitors. It implies that distribution
propagation differs from typical label propagation, which in
turn demonstrates the necessity of our method when counts
or frequencies are used as features. Figure 5(c) shows that
the ranking performance only changes slightly though µ differs significantly. This verifies one of our claims that the proposed method is not sensitive to the scale of parameter µ.
Conclusion
In this paper we describe a method to propagate ranking
functions on a graph and present an algorithm to perform
distribution propagation. The method is used to address the
issue caused by limited and imbalanced training data to
propagate thousands of ranking functions on the same set of
data. The proposed propagation approach is centered around
distribution data and allows for the interactions between
neighboring histogram bins. We derive a closed-form solution for the optimization, which makes our algorithm easy to
implement. The experimental result on real world problems
demonstrates the usefulness of our propagation method.
Result and discussion. We select 100 users in each random trial, and train a strong ranking function for each of
them using all available ratings. These strong functions are
used as the initial ranking functions to propagate, and then
additional 10,000 randomly selected user ratings are added
to the training data step by step, which are used either to train
unknown ranking functions, or to improve the weak ones. At
1
Average NDCG w.r.t. different µ
Figure 5: Evaluation on movie posters (standard deviation is denoted by shade).
http://www.boredpanda.com/movie-poster-cliches/
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The authors gratefully acknowledge support of this research
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and NSF Grant NSF IIS-0801528.
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